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The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

In geometry, an epicycloid or hypercycloid is a plane curve produced by tracing the path of a chosen point on the circumference of a circle—called an epicycle—which rolls without slipping around a fixed circle. It is a particular kind of roulette.



If the smaller circle has radius r, and the larger circle has radius R = kr, then the parametric equations for the curve can be given by either:




(Assuming the initial point lies on the larger circle.)

If k is an integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable).

If k is a rational number, say k=p/q expressed in simplest terms, then the curve has p cusps.

If k is an irrational number, then the curve never closes, and forms a dense subset of the space between the larger circle and a circle of radius R + 2r.

When measured in radian,   takes value from   to  where LCM is least common multiple.

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid, two cusps is a nephroid.

An epicycloid and its evolute are similar.[1]


sketch for proof

We assume that the position of   is what we want to solve,   is the radian from the tangential point to the moving point  , and   is the radian from the starting point to the tangential point.

Since there is no sliding between the two cycles, then we have that


By the definition of radian (which is the rate arc over radius), then we have that


From these two conditions, we get the identity


By calculating, we get the relation between   and  , which is


From the figure, we see the position of the point   clearly.


See alsoEdit


  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161, 168–170, 175. ISBN 978-0-486-60288-2.

External linksEdit